Additive white Gaussian noise ( AWGN ) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
99-503: Wideband noise comes from many natural noise sources, such as the thermal vibrations of atoms in conductors (referred to as thermal noise or Johnson–Nyquist noise ), shot noise , black-body radiation from the earth and other warm objects, and from celestial sources such as the Sun. The central limit theorem of probability theory indicates that the summation of many random processes will tend to have distribution called Gaussian or Normal. AWGN
198-496: A frequency band of bandwidth Δ f {\displaystyle \Delta f} (Figure 3) has a mean square voltage of: where k B {\displaystyle k_{\rm {B}}} is the Boltzmann constant ( 1.380 649 × 10 joules per kelvin ). While this equation applies to ideal resistors (i.e. pure resistances without any frequency-dependence) at non-extreme frequency and temperatures,
297-461: A quantum Hall resistor , held at the triple-point temperature of water . The voltage is measured over a period of 100 days and integrated. This was done in 2017, when the triple point of water's temperature was 273.16 K by definition, and the Boltzmann constant was experimentally measurable. Because the acoustic gas thermometry reached 0.2 ppm in uncertainty, and Johnson noise 2.8 ppm, this fulfilled
396-522: A capacitor can be derived from this relationship, without consideration of resistance. The Johnson–Nyquist noise has applications in precision measurements, in which it is typically called "Johnson noise thermometry". For example, the NIST in 2017 used the Johnson noise thermometry to measure the Boltzmann constant with uncertainty less than 3 ppm . It accomplished this by using Josephson voltage standard and
495-497: A channel remains open, but it can be upper bounded by another important graph invariant, the Lovász number . The noisy-channel coding theorem states that for any error probability ε > 0 and for any transmission rate R less than the channel capacity C , there is an encoding and decoding scheme transmitting data at rate R whose error probability is less than ε, for a sufficiently large block length. Also, for any rate greater than
594-515: A coherent carrier signal. The instantaneous response of the noise vector cannot be precisely predicted, however, its time-averaged response can be statistically predicted. As shown in the graph, we confidently predict that the noise phasor will reside about 38% of the time inside the 1 σ circle, about 86% of the time inside the 2 σ circle, and about 98% of the time inside the 3 σ circle. Johnson%E2%80%93Nyquist noise Johnson–Nyquist noise ( thermal noise , Johnson noise , or Nyquist noise )
693-647: A combined manner provides the same theoretical capacity as using them independently. More formally, let p 1 {\displaystyle p_{1}} and p 2 {\displaystyle p_{2}} be two independent channels modelled as above; p 1 {\displaystyle p_{1}} having an input alphabet X 1 {\displaystyle {\mathcal {X}}_{1}} and an output alphabet Y 1 {\displaystyle {\mathcal {Y}}_{1}} . Idem for p 2 {\displaystyle p_{2}} . We define
792-544: A concave (downward) function of x , to get: Because each codeword individually satisfies the power constraint, the average also satisfies the power constraint. Therefore, which we may apply to simplify the inequality above and get: Therefore, it must be that R ≤ 1 2 log ( 1 + P N ) + ε n {\displaystyle R\leq {\frac {1}{2}}\log \left(1+{\frac {P}{N}}\right)+\varepsilon _{n}} . Therefore, R must be less than
891-456: A cooperative framework inspired by generative adversarial networks . CORTICAL consists of two cooperative networks: a generator with the objective of learning to sample from the capacity-achieving input distribution, and a discriminator with the objective to learn to distinguish between paired and unpaired channel input-output samples and estimates I ( X ; Y ) {\displaystyle I(X;Y)} . This section focuses on
990-1575: A distribution p X 1 , X 2 {\displaystyle p_{X_{1},X_{2}}} such that I ( X 1 , X 2 : Y 1 , Y 2 ) ≥ I ( X 1 : Y 1 ) + I ( X 2 : Y 2 ) {\displaystyle I(X_{1},X_{2}:Y_{1},Y_{2})\geq I(X_{1}:Y_{1})+I(X_{2}:Y_{2})} . In fact, π 1 {\displaystyle \pi _{1}} and π 2 {\displaystyle \pi _{2}} , two probability distributions for X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} achieving C ( p 1 ) {\displaystyle C(p_{1})} and C ( p 2 ) {\displaystyle C(p_{2})} , suffice: ie. C ( p 1 × p 2 ) ≥ C ( p 1 ) + C ( p 2 ) {\displaystyle C(p_{1}\times p_{2})\geq C(p_{1})+C(p_{2})} Now let us show that C ( p 1 × p 2 ) ≤ C ( p 1 ) + C ( p 2 ) {\displaystyle C(p_{1}\times p_{2})\leq C(p_{1})+C(p_{2})} . Let π 12 {\displaystyle \pi _{12}} be some distribution for
1089-602: A generalized expression that applies to non-equal and complex impedances too. And while Nyquist above used k B T {\displaystyle k_{\rm {B}}T} according to classical theory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated the Planck constant h {\displaystyle h} (from the new theory of quantum mechanics ). The 4 k B T R {\displaystyle 4k_{\text{B}}TR} voltage noise described above
SECTION 10
#17330851602001188-3816: A given pair ( x 1 , x 2 ) {\displaystyle (x_{1},x_{2})} , we can rewrite H ( Y 1 , Y 2 | X 1 , X 2 = x 1 , x 2 ) {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2}=x_{1},x_{2})} as: H ( Y 1 , Y 2 | X 1 , X 2 = x 1 , x 2 ) = ∑ ( y 1 , y 2 ) ∈ Y 1 × Y 2 P ( Y 1 , Y 2 = y 1 , y 2 | X 1 , X 2 = x 1 , x 2 ) log ( P ( Y 1 , Y 2 = y 1 , y 2 | X 1 , X 2 = x 1 , x 2 ) ) = ∑ ( y 1 , y 2 ) ∈ Y 1 × Y 2 P ( Y 1 , Y 2 = y 1 , y 2 | X 1 , X 2 = x 1 , x 2 ) [ log ( P ( Y 1 = y 1 | X 1 = x 1 ) ) + log ( P ( Y 2 = y 2 | X 2 = x 2 ) ) ] = H ( Y 1 | X 1 = x 1 ) + H ( Y 2 | X 2 = x 2 ) {\displaystyle {\begin{aligned}H(Y_{1},Y_{2}|X_{1},X_{2}=x_{1},x_{2})&=\sum _{(y_{1},y_{2})\in {\mathcal {Y}}_{1}\times {\mathcal {Y}}_{2}}\mathbb {P} (Y_{1},Y_{2}=y_{1},y_{2}|X_{1},X_{2}=x_{1},x_{2})\log(\mathbb {P} (Y_{1},Y_{2}=y_{1},y_{2}|X_{1},X_{2}=x_{1},x_{2}))\\&=\sum _{(y_{1},y_{2})\in {\mathcal {Y}}_{1}\times {\mathcal {Y}}_{2}}\mathbb {P} (Y_{1},Y_{2}=y_{1},y_{2}|X_{1},X_{2}=x_{1},x_{2})[\log(\mathbb {P} (Y_{1}=y_{1}|X_{1}=x_{1}))+\log(\mathbb {P} (Y_{2}=y_{2}|X_{2}=x_{2}))]\\&=H(Y_{1}|X_{1}=x_{1})+H(Y_{2}|X_{2}=x_{2})\end{aligned}}} By summing this equality over all ( x 1 , x 2 ) {\displaystyle (x_{1},x_{2})} , we obtain H ( Y 1 , Y 2 | X 1 , X 2 ) = H ( Y 1 | X 1 ) + H ( Y 2 | X 2 ) {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2})=H(Y_{1}|X_{1})+H(Y_{2}|X_{2})} . We can now give an upper bound over mutual information: I ( X 1 , X 2 : Y 1 , Y 2 ) ≤ H ( Y 1 ) + H ( Y 2 ) − H ( Y 1 | X 1 ) − H ( Y 2 | X 2 ) = I ( X 1 : Y 1 ) + I ( X 2 : Y 2 ) {\displaystyle {\begin{aligned}I(X_{1},X_{2}:Y_{1},Y_{2})&\leq H(Y_{1})+H(Y_{2})-H(Y_{1}|X_{1})-H(Y_{2}|X_{2})\\&=I(X_{1}:Y_{1})+I(X_{2}:Y_{2})\end{aligned}}} This relation
1287-506: A hot resistor will create electromagnetic waves on a transmission line just as a hot object will create electromagnetic waves in free space. In 1946, Robert H. Dicke elaborated on the relationship, and further connected it to properties of antennas, particularly the fact that the average antenna aperture over all different directions cannot be larger than λ 2 4 π {\displaystyle {\tfrac {\lambda ^{2}}{4\pi }}} , where λ
1386-470: A maximum energy of n ( P + N ) {\displaystyle n(P+N)} and therefore must occupy a sphere of radius n ( P + N ) {\textstyle {\sqrt {n(P+N)}}} . Each codeword sphere has radius n N {\displaystyle {\sqrt {nN}}} . The volume of an n -dimensional sphere is directly proportional to r n {\displaystyle r^{n}} , so
1485-776: A more accurate general form accounts for complex impedances and quantum effects. Conventional electronics generally operate over a more limited bandwidth , so Johnson's equation is often satisfactory. The mean square voltage per hertz of bandwidth is 4 k B T R {\displaystyle 4k_{\text{B}}TR} and may be called the power spectral density (Figure 2). Its square root at room temperature (around 300 K) approximates to 0.13 R {\displaystyle {\sqrt {R}}} in units of nanovolts / √ hertz . A 10 kΩ resistor, for example, would have approximately 13 nanovolts / √ hertz at room temperature. The square root of
1584-496: A noise-free resistor in parallel with a gaussian noise current source with the following RMS current: Ideal capacitors , as lossless devices, do not have thermal noise. However, the combination of a resistor and a capacitor (an RC circuit , a common low-pass filter ) has what is called kTC noise. The noise bandwidth of an RC circuit is Δ f = 1 4 R C . {\displaystyle \Delta f{=}{\tfrac {1}{4RC}}.} When this
1683-420: A non-zero probability that the channel is in deep fade, the capacity of the slow-fading channel in strict sense is zero. However, it is possible to determine the largest value of R {\displaystyle R} such that the outage probability p o u t {\displaystyle p_{out}} is less than ϵ {\displaystyle \epsilon } . This value
1782-1174: A random variable corresponding to the output of X 1 {\displaystyle X_{1}} through the channel p 1 {\displaystyle p_{1}} , and Y 2 {\displaystyle Y_{2}} for X 2 {\displaystyle X_{2}} through p 2 {\displaystyle p_{2}} . By definition C ( p 1 × p 2 ) = sup p X 1 , X 2 ( I ( X 1 , X 2 : Y 1 , Y 2 ) ) {\displaystyle C(p_{1}\times p_{2})=\sup _{p_{X_{1},X_{2}}}(I(X_{1},X_{2}:Y_{1},Y_{2}))} . Since X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are independent, as well as p 1 {\displaystyle p_{1}} and p 2 {\displaystyle p_{2}} , ( X 1 , Y 1 ) {\displaystyle (X_{1},Y_{1})}
1881-454: A resistor R S {\displaystyle R_{\text{S}}} can transfer to the remaining circuit. The maximum power transfer happens when the Thévenin equivalent resistance R L {\displaystyle R_{\rm {L}}} of the remaining circuit matches R S {\displaystyle R_{\text{S}}} . In this case, each of
1980-414: A set of cross-spectral density functions relating the different noise voltages, where the Z m n {\displaystyle Z_{mn}} are the elements of the impedance matrix Z {\displaystyle \mathbf {Z} } . Again, an alternative description of the noise is instead in terms of parallel current sources applied at each port. Their cross-spectral density
2079-459: A value arbitrarily close to the capacity derived earlier, as ε n → 0 {\displaystyle \varepsilon _{n}\rightarrow 0} . In serial data communications, the AWGN mathematical model is used to model the timing error caused by random jitter (RJ). The graph to the right shows an example of timing errors associated with AWGN. The variable Δ t represents
SECTION 20
#17330851602002178-446: Is independent and identically distributed and drawn from a zero-mean normal distribution with variance N {\displaystyle N} (the noise). The Z i {\displaystyle Z_{i}} are further assumed to not be correlated with the X i {\displaystyle X_{i}} . The capacity of the channel is infinite unless the noise N {\displaystyle N}
2277-409: Is a code of rate R arbitrarily close to the capacity derived earlier. Here we show that rates above the capacity C = 1 2 log ( 1 + P N ) {\displaystyle C={\frac {1}{2}}\log \left(1+{\frac {P}{N}}\right)} are not achievable. Suppose that the power constraint is satisfied for a codebook, and further suppose that
2376-410: Is a special case for a purely resistive component for low to moderate frequencies. In general, the thermal electrical noise continues to be related to resistive response in many more generalized electrical cases, as a consequence of the fluctuation-dissipation theorem . Below a variety of generalizations are noted. All of these generalizations share a common limitation, that they only apply in cases where
2475-516: Is approximately white , meaning that its power spectral density is nearly constant throughout the frequency spectrum (Figure 2). When limited to a finite bandwidth and viewed in the time domain (as sketched in Figure 1), thermal noise has a nearly Gaussian amplitude distribution . For the general case, this definition applies to charge carriers in any type of conducting medium (e.g. ions in an electrolyte ), not just resistors . Thermal noise
2574-492: Is distinct from shot noise , which consists of additional current fluctuations that occur when a voltage is applied and a macroscopic current starts to flow. In 1905, in one of Albert Einstein 's Annus mirabilis papers the theory of Brownian motion was first solved in terms of thermal fluctuations. The following year, in a second paper about Brownian motion, Einstein suggested that the same phenomena could be applied to derive thermally-agitated currents, but did not carry out
2673-521: Is given by where Y = Z − 1 {\displaystyle \mathbf {Y} =\mathbf {Z} ^{-1}} is the admittance matrix . [REDACTED] This article incorporates public domain material from Federal Standard 1037C . General Services Administration . Archived from the original on 2022-01-22. (in support of MIL-STD-188 ). Channel capacity Channel capacity , in electrical engineering , computer science , and information theory ,
2772-534: Is independent of ( X 2 , Y 2 ) {\displaystyle (X_{2},Y_{2})} . We can apply the following property of mutual information : I ( X 1 , X 2 : Y 1 , Y 2 ) = I ( X 1 : Y 1 ) + I ( X 2 : Y 2 ) {\displaystyle I(X_{1},X_{2}:Y_{1},Y_{2})=I(X_{1}:Y_{1})+I(X_{2}:Y_{2})} For now we only need to find
2871-641: Is known as the ϵ {\displaystyle \epsilon } -outage capacity. In a fast-fading channel , where the latency requirement is greater than the coherence time and the codeword length spans many coherence periods, one can average over many independent channel fades by coding over a large number of coherence time intervals. Thus, it is possible to achieve a reliable rate of communication of E ( log 2 ( 1 + | h | 2 S N R ) ) {\displaystyle \mathbb {E} (\log _{2}(1+|h|^{2}SNR))} [bits/s/Hz] and it
2970-452: Is logarithmic in power and approximately linear in bandwidth. This is called the bandwidth-limited regime . When the SNR is small (SNR ≪ 0 dB), the capacity C ≈ P ¯ N 0 ln 2 {\displaystyle C\approx {\frac {\bar {P}}{N_{0}\ln 2}}} is linear in power but insensitive to bandwidth. This is called
3069-489: Is maximized when: Thus the channel capacity C {\displaystyle C} for the AWGN channel is given by: Suppose that we are sending messages through the channel with index ranging from 1 {\displaystyle 1} to M {\displaystyle M} , the number of distinct possible messages. If we encode the M {\displaystyle M} messages to n {\displaystyle n} bits, then we define
Additive white Gaussian noise - Misplaced Pages Continue
3168-463: Is meaningful to speak of this value as the capacity of the fast-fading channel. Feedback capacity is the greatest rate at which information can be reliably transmitted, per unit time, over a point-to-point communication channel in which the receiver feeds back the channel outputs to the transmitter. Information-theoretic analysis of communication systems that incorporate feedback is more complicated and challenging than without feedback. Possibly, this
3267-484: Is nonzero, and the X i {\displaystyle X_{i}} are sufficiently constrained. The most common constraint on the input is the so-called "power" constraint, requiring that for a codeword ( x 1 , x 2 , … , x k ) {\displaystyle (x_{1},x_{2},\dots ,x_{k})} transmitted through the channel, we have: where P {\displaystyle P} represents
3366-463: Is often a limiting noise source, for example in image sensors . Any system in thermal equilibrium has state variables with a mean energy of kT / 2 per degree of freedom . Using the formula for energy on a capacitor ( E = 1 / 2 CV ), mean noise energy on a capacitor can be seen to also be 1 / 2 C kT / C = kT / 2 . Thermal noise on
3465-479: Is often used as a channel model in which the only impairment to communication is a linear addition of wideband or white noise with a constant spectral density (expressed as watts per hertz of bandwidth ) and a Gaussian distribution of amplitude. The model does not account for fading , frequency selectivity, interference , nonlinearity or dispersion . However, it produces simple and tractable mathematical models which are useful for gaining insight into
3564-476: Is preserved at the supremum. Therefore Combining the two inequalities we proved, we obtain the result of the theorem: If G is an undirected graph , it can be used to define a communications channel in which the symbols are the graph vertices, and two codewords may be confused with each other if their symbols in each position are equal or adjacent. The computational complexity of finding the Shannon capacity of such
3663-442: Is proportional to absolute temperature , so some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to improve their signal-to-noise ratio . The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem , where generalized impedance or generalized susceptibility is used to characterize the medium. Thermal noise in an ideal resistor
3762-462: Is represented by a series of outputs Y i {\displaystyle Y_{i}} at discrete-time event index i {\displaystyle i} . Y i {\displaystyle Y_{i}} is the sum of the input X i {\displaystyle X_{i}} and noise, Z i {\displaystyle Z_{i}} , where Z i {\displaystyle Z_{i}}
3861-399: Is substituted into the thermal noise equation, the result has an unusually simple form as the value of the resistance ( R ) drops out of the equation. This is because higher R decreases the bandwidth as much as it increases the noise. The mean-square and RMS noise voltage generated in such a filter are: The noise charge Q n {\displaystyle Q_{n}} is
3960-486: Is the electrical admittance ; note that Re [ Y ( f ) ] = Re [ Z ( f ) ] | Z ( f ) | 2 . {\displaystyle \operatorname {Re} [Y(f)]{=}{\tfrac {\operatorname {Re} [Z(f)]}{|Z(f)|^{2}}}\,.} With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near absolute zero ),
4059-459: Is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons ) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage . Thermal noise is present in all electrical circuits , and in sensitive electronic equipment (such as radio receivers ) can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise
Additive white Gaussian noise - Misplaced Pages Continue
4158-553: Is the gain of subchannel n {\displaystyle n} , with λ {\displaystyle \lambda } chosen to meet the power constraint. In a slow-fading channel , where the coherence time is greater than the latency requirement, there is no definite capacity as the maximum rate of reliable communications supported by the channel, log 2 ( 1 + | h | 2 S N R ) {\displaystyle \log _{2}(1+|h|^{2}SNR)} , depends on
4257-524: Is the square of this current multiplied by R 2 {\displaystyle R_{2}} , which simplifies to: Setting this P 1 {\textstyle P_{\text{1}}} equal to the earlier average power expression P 1 ¯ {\textstyle {\overline {P_{1}}}} allows solving for the average of V 1 2 {\textstyle V_{1}^{2}} over that bandwidth: Nyquist used similar reasoning to provide
4356-430: Is the theoretical maximum rate at which information can be reliably transmitted over a communication channel . Following the terms of the noisy-channel coding theorem , the channel capacity of a given channel is the highest information rate (in units of information per unit time) that can be achieved with arbitrarily small error probability. Information theory , developed by Claude E. Shannon in 1948, defines
4455-497: Is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch . Though an ideal switch's open resistance is infinite, the formula still applies. However, now the RMS voltage must be interpreted not as a time average, but as an average over many such reset events, since the voltage is constant when the bandwidth is zero. In this sense, the Johnson noise of an RC circuit can be seen to be inherent, an effect of
4554-499: Is wavelength. This comes from the different frequency dependence of 3D versus 1D Planck's law. Richard Q. Twiss extended Nyquist's formulas to multi- port passive electrical networks, including non-reciprocal devices such as circulators and isolators . Thermal noise appears at every port, and can be described as random series voltage sources in series with each port. The random voltages at different ports may be correlated, and their amplitudes and correlations are fully described by
4653-479: The capacitance times the voltage: This charge noise is the origin of the term " kTC noise". Although independent of the resistor's value, 100% of the kTC noise arises in the resistor. Therefore, it would incorrect to double-count both a resistor's thermal noise and its associated kTC noise, and the temperature of the resistor alone should be used, even if the resistor and the capacitor are at different temperatures. Some values are tabulated below: An extreme case
4752-506: The conditional probability distribution function of Y {\displaystyle Y} given X {\displaystyle X} , which is an inherent fixed property of the communication channel. Then the choice of the marginal distribution p X ( x ) {\displaystyle p_{X}(x)} completely determines the joint distribution p X , Y ( x , y ) {\displaystyle p_{X,Y}(x,y)} due to
4851-493: The differential entropy of a Gaussian gives: Because X {\displaystyle X} and Z {\displaystyle Z} are independent and their sum gives Y {\displaystyle Y} : From this bound, we infer from a property of the differential entropy that Therefore, the channel capacity is given by the highest achievable bound on the mutual information : Where I ( X ; Y ) {\displaystyle I(X;Y)}
4950-476: The natural logarithm is used, assuming B is in hertz ; the signal and noise powers S and N are expressed in a linear power unit (like watts or volts ). Since S/N figures are often cited in dB , a conversion may be needed. For example, a signal-to-noise ratio of 30 dB corresponds to a linear power ratio of 10 30 / 10 = 10 3 = 1000 {\displaystyle 10^{30/10}=10^{3}=1000} . To determine
5049-465: The phasor domain, statistical analysis reveals that the amplitudes of the real and imaginary contributions are independent variables which follow the Gaussian distribution model. When combined, the resultant phasor's magnitude is a Rayleigh-distributed random variable, while the phase is uniformly distributed from 0 to 2 π . The graph to the right shows an example of how bandlimited AWGN can affect
SECTION 50
#17330851602005148-423: The power spectral density of the series noise voltage is The function η ( f ) {\displaystyle \eta (f)} is approximately 1, except at very high frequencies or near absolute zero (see below). The real part of impedance, Re [ Z ( f ) ] {\displaystyle \operatorname {Re} [Z(f)]} , is in general frequency dependent and so
5247-740: The power-limited regime . The bandwidth-limited regime and power-limited regime are illustrated in the figure. The capacity of the frequency-selective channel is given by so-called water filling power allocation, where P n ∗ = max { ( 1 λ − N 0 | h ¯ n | 2 ) , 0 } {\displaystyle P_{n}^{*}=\max \left\{\left({\frac {1}{\lambda }}-{\frac {N_{0}}{|{\bar {h}}_{n}|^{2}}}\right),0\right\}} and | h ¯ n | 2 {\displaystyle |{\bar {h}}_{n}|^{2}}
5346-602: The shot noise . Frits Zernike working in electrical metrology, found unusual random deflections while working with high-sensitive galvanometers . He rejected the idea that the noise was mechanical, and concluded that it was of thermal nature. In 1927, he introduced the idea of autocorrelations to electrical measurements and calculated the time detection limit. His work coincided with de Haas-Lorentz' prediction. The same year, working independently without any knowledge of Zernike's work, John B. Johnson working in Bell Labs found
5445-571: The AWGN channel capacity is where P ¯ N 0 W {\displaystyle {\frac {\bar {P}}{N_{0}W}}} is the received signal-to-noise ratio (SNR). This result is known as the Shannon–Hartley theorem . When the SNR is large (SNR ≫ 0 dB), the capacity C ≈ W log 2 P ¯ N 0 W {\displaystyle C\approx W\log _{2}{\frac {\bar {P}}{N_{0}W}}}
5544-465: The AWGN channel with noise level N {\displaystyle N} . When received, the codeword vector variance is now N {\displaystyle N} , and its mean is the codeword sent. The vector is very likely to be contained in a sphere of radius n ( N + ε ) {\textstyle {\sqrt {n(N+\varepsilon )}}} around the codeword sent. If we decode by mapping every message received onto
5643-617: The Johnson–Nyquist noise is not white noise. The RMS noise voltage over a span of frequencies f 1 {\displaystyle f_{1}} to f 2 {\displaystyle f_{2}} can be found by taking the square root of integration of the power spectral density: Alternatively, a parallel noise current can be used to describe Johnson noise, its power spectral density being where Y ( f ) = 1 Z ( f ) {\displaystyle Y(f){=}{\tfrac {1}{Z(f)}}}
5742-507: The advent of novel error correction coding mechanisms that have resulted in achieving performance very close to the limits promised by channel capacity. The basic mathematical model for a communication system is the following: where: Let X {\displaystyle X} and Y {\displaystyle Y} be modeled as random variables. Furthermore, let p Y | X ( y | x ) {\displaystyle p_{Y|X}(y|x)} be
5841-1208: The alphabet of X 1 {\displaystyle X_{1}} , Y 1 {\displaystyle {\mathcal {Y}}_{1}} for Y 1 {\displaystyle Y_{1}} , and analogously X 2 {\displaystyle {\mathcal {X}}_{2}} and Y 2 {\displaystyle {\mathcal {Y}}_{2}} . By definition of mutual information, we have I ( X 1 , X 2 : Y 1 , Y 2 ) = H ( Y 1 , Y 2 ) − H ( Y 1 , Y 2 | X 1 , X 2 ) ≤ H ( Y 1 ) + H ( Y 2 ) − H ( Y 1 , Y 2 | X 1 , X 2 ) {\displaystyle {\begin{aligned}I(X_{1},X_{2}:Y_{1},Y_{2})&=H(Y_{1},Y_{2})-H(Y_{1},Y_{2}|X_{1},X_{2})\\&\leq H(Y_{1})+H(Y_{2})-H(Y_{1},Y_{2}|X_{1},X_{2})\end{aligned}}} Let us rewrite
5940-522: The available noise power can be easily approximated as 10 log 10 ( Δ f ) − 173.8 {\displaystyle 10\ \log _{10}(\Delta f)-173.8} in dBm for a bandwidth in hertz. Some example available noise power in dBm are tabulated below: Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors" used concepts about potential energy and harmonic oscillators from
6039-465: The calculation as he considered it to be untestable. Geertruida de Haas-Lorentz , daughter of Hendrik Lorentz , in her doctoral thesis of 1912, expanded on Einstein stochastic theory and first applied it to the study of electrons, deriving a formula for the mean-squared value of the thermal current. Walter H. Schottky studied the problem in 1918, while studying thermal noise using Einstein's theories, experimentally discovered another kind of noise,
SECTION 60
#17330851602006138-455: The channel p 1 × p 2 {\displaystyle p_{1}\times p_{2}} defining ( X 1 , X 2 ) {\displaystyle (X_{1},X_{2})} and the corresponding output ( Y 1 , Y 2 ) {\displaystyle (Y_{1},Y_{2})} . Let X 1 {\displaystyle {\mathcal {X}}_{1}} be
6237-440: The channel capacity, it is necessary to find the capacity-achieving distribution p X ( x ) {\displaystyle p_{X}(x)} and evaluate the mutual information I ( X ; Y ) {\displaystyle I(X;Y)} . Research has mostly focused on studying additive noise channels under certain power constraints and noise distributions, as analytical methods are not feasible in
6336-408: The channel capacity, the probability of error at the receiver goes to 0.5 as the block length goes to infinity. An application of the channel capacity concept to an additive white Gaussian noise (AWGN) channel with B Hz bandwidth and signal-to-noise ratio S/N is the Shannon–Hartley theorem : C is measured in bits per second if the logarithm is taken in base 2, or nats per second if
6435-546: The codeword at the center of this sphere, then an error occurs only when the received vector is outside of this sphere, which is very unlikely. Each codeword vector has an associated sphere of received codeword vectors which are decoded to it and each such sphere must map uniquely onto a codeword. Because these spheres therefore must not intersect, we are faced with the problem of sphere packing . How many distinct codewords can we pack into our n {\displaystyle n} -bit codeword vector? The received vectors have
6534-447: The combined resistance is I 1 = V 1 R 1 + R 2 = V 1 2 R 1 {\textstyle I_{1}{=}{\tfrac {V_{1}}{R_{1}+R_{2}}}{=}{\tfrac {V_{1}}{2R_{1}}}} , so the power transferred from R 1 {\displaystyle R_{1}} to R 2 {\displaystyle R_{2}}
6633-413: The electrical component under consideration is purely passive and linear. Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors. Such a component can be described by a frequency-dependent complex electrical impedance Z ( f ) {\displaystyle Z(f)} . The formula for
6732-408: The encoded message of codeword index i . Then: Let P i {\displaystyle P_{i}} be the average power of the codeword of index i: where the sum is over all input messages w {\displaystyle w} . X i {\displaystyle X_{i}} and Z i {\displaystyle Z_{i}} are independent, thus
6831-409: The equipartition law of Boltzmann and Maxwell to explain Johnson's experimental result. Nyquist's thought experiment summed the energy contribution of each standing wave mode of oscillation on a long lossless transmission line between two equal resistors ( R 1 = R 2 {\displaystyle R_{1}{=}R_{2}} ). According to the conclusion of Figure 5,
6930-416: The expectation of the power of Y i {\displaystyle Y_{i}} is, for noise level N {\displaystyle N} : And, if Y i {\displaystyle Y_{i}} is normally distributed, we have that Therefore, We may apply Jensen's equality to log ( 1 + x ) {\displaystyle \log(1+x)} ,
7029-425: The identity which, in turn, induces a mutual information I ( X ; Y ) {\displaystyle I(X;Y)} . The channel capacity is defined as where the supremum is taken over all possible choices of p X ( x ) {\displaystyle p_{X}(x)} . Channel capacity is additive over independent channels. It means that using two independent channels in
7128-895: The joint Asymptotic Equipartition Property the same applies to P ( V ) {\displaystyle P(V)} . Therefore, for a sufficiently large n {\displaystyle n} , both P ( U ) {\displaystyle P(U)} and P ( V ) {\displaystyle P(V)} are each less than ε {\displaystyle \varepsilon } . Since X n ( i ) {\displaystyle X^{n}(i)} and X n ( j ) {\displaystyle X^{n}(j)} are independent for i ≠ j {\displaystyle i\neq j} , we have that X n ( i ) {\displaystyle X^{n}(i)} and Y n {\displaystyle Y^{n}} are also independent. Therefore, by
7227-689: The joint AEP, P ( E j ) = 2 − n ( I ( X ; Y ) − 3 ε ) {\displaystyle P(E_{j})=2^{-n(I(X;Y)-3\varepsilon )}} . This allows us to calculate P e ( n ) {\displaystyle P_{e}^{(n)}} , the probability of error as follows: Therefore, as n approaches infinity, P e ( n ) {\displaystyle P_{e}^{(n)}} goes to zero and R < I ( X ; Y ) − 3 ε {\displaystyle R<I(X;Y)-3\varepsilon } . Therefore, there
7326-777: The last term of entropy . H ( Y 1 , Y 2 | X 1 , X 2 ) = ∑ ( x 1 , x 2 ) ∈ X 1 × X 2 P ( X 1 , X 2 = x 1 , x 2 ) H ( Y 1 , Y 2 | X 1 , X 2 = x 1 , x 2 ) {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2})=\sum _{(x_{1},x_{2})\in {\mathcal {X}}_{1}\times {\mathcal {X}}_{2}}\mathbb {P} (X_{1},X_{2}=x_{1},x_{2})H(Y_{1},Y_{2}|X_{1},X_{2}=x_{1},x_{2})} By definition of
7425-563: The majority of other scenarios. Hence, alternative approaches such as, investigation on the input support, relaxations and capacity bounds, have been proposed in the literature. The capacity of a discrete memoryless channel can be computed using the Blahut-Arimoto algorithm . Deep learning can be used to estimate the channel capacity. In fact, the channel capacity and the capacity-achieving distribution of any discrete-time continuous memoryless vector channel can be obtained using CORTICAL,
7524-509: The maximum channel power. Therefore, the channel capacity for the power-constrained channel is given by: where f {\displaystyle f} is the distribution of X {\displaystyle X} . Expand I ( X ; Y ) {\displaystyle I(X;Y)} , writing it in terms of the differential entropy : But X {\displaystyle X} and Z {\displaystyle Z} are independent, therefore: Evaluating
7623-405: The maximum number of uniquely decodeable spheres that can be packed into our sphere with transmission power P is: By this argument, the rate R can be no more than 1 2 log ( 1 + P N ) {\displaystyle {\frac {1}{2}}\log \left(1+{\frac {P}{N}}\right)} . In this section, we show achievability of the upper bound on
7722-764: The mean square voltage yields the root mean square (RMS) voltage observed over the bandwidth Δ f {\displaystyle \Delta f} : A resistor with thermal noise can be represented by its Thévenin equivalent circuit (Figure 4B) consisting of a noiseless resistor in series with a gaussian noise voltage source with the above RMS voltage. Around room temperature, 3 kΩ provides almost one microvolt of RMS noise over 20 kHz (the human hearing range ) and 60 Ω·Hz for R Δ f {\displaystyle R\,\Delta f} corresponds to almost one nanovolt of RMS noise. A resistor with thermal noise can also be converted into its Norton equivalent circuit (Figure 4C) consisting of
7821-1138: The messages follow a uniform distribution. Let W {\displaystyle W} be the input messages and W ^ {\displaystyle {\hat {W}}} the output messages. Thus the information flows as: W ⟶ X ( n ) ( W ) ⟶ Y ( n ) ⟶ W ^ {\displaystyle W\longrightarrow X^{(n)}(W)\longrightarrow Y^{(n)}\longrightarrow {\hat {W}}} Making use of Fano's inequality gives: H ( W ∣ W ^ ) ≤ 1 + n R P e ( n ) = n ε n {\displaystyle H(W\mid {\hat {W}})\leq 1+nRP_{e}^{(n)}=n\varepsilon _{n}} where ε n → 0 {\displaystyle \varepsilon _{n}\rightarrow 0} as P e ( n ) → 0 {\displaystyle P_{e}^{(n)}\rightarrow 0} Let X i {\displaystyle X_{i}} be
7920-486: The multiplying factor η ( f ) {\displaystyle \eta (f)} mentioned earlier is in general given by: At very high frequencies ( f ≳ k B T h {\displaystyle f\gtrsim {\tfrac {k_{\text{B}}T}{h}}} ), the function η ( f ) {\displaystyle \eta (f)} starts to exponentially decrease to zero. At room temperature this transition occurs in
8019-460: The notion of channel capacity and provides a mathematical model by which it may be computed. The key result states that the capacity of the channel, as defined above, is given by the maximum of the mutual information between the input and output of the channel, where the maximization is with respect to the input distribution. The notion of channel capacity has been central to the development of modern wireline and wireless communication systems, with
8118-455: The output. The directed information was coined by James Massey in 1990, who showed that its an upper bound on feedback capacity. For memoryless channels , Shannon showed that feedback does not increase the capacity, and the feedback capacity coincides with the channel capacity characterized by the mutual information between the input and the output. The feedback capacity is known as a closed-form expression only for several examples such as:
8217-482: The power constraint probabilistically. Received messages are decoded to a message in the codebook which is uniquely jointly typical. If there is no such message or if the power constraint is violated, a decoding error is declared. Let X n ( i ) {\displaystyle X^{n}(i)} denote the codeword for message i {\displaystyle i} , while Y n {\displaystyle Y^{n}} is, as before
8316-441: The preconditions for a redefinition. After the 2019 redefinition , the kelvin was defined so that the Boltzmann constant is 1.380649×10 J⋅K , and the triple point of water became experimentally measurable. Inductors are the dual of capacitors. Analogous to kTC noise, a resistor with an inductor L {\displaystyle L} results in a noise current that is independent of resistance: The noise generated at
8415-1734: The product channel p 1 × p 2 {\displaystyle p_{1}\times p_{2}} as ∀ ( x 1 , x 2 ) ∈ ( X 1 , X 2 ) , ( y 1 , y 2 ) ∈ ( Y 1 , Y 2 ) , ( p 1 × p 2 ) ( ( y 1 , y 2 ) | ( x 1 , x 2 ) ) = p 1 ( y 1 | x 1 ) p 2 ( y 2 | x 2 ) {\displaystyle \forall (x_{1},x_{2})\in ({\mathcal {X}}_{1},{\mathcal {X}}_{2}),\;(y_{1},y_{2})\in ({\mathcal {Y}}_{1},{\mathcal {Y}}_{2}),\;(p_{1}\times p_{2})((y_{1},y_{2})|(x_{1},x_{2}))=p_{1}(y_{1}|x_{1})p_{2}(y_{2}|x_{2})} This theorem states: C ( p 1 × p 2 ) = C ( p 1 ) + C ( p 2 ) {\displaystyle C(p_{1}\times p_{2})=C(p_{1})+C(p_{2})} We first show that C ( p 1 × p 2 ) ≥ C ( p 1 ) + C ( p 2 ) {\displaystyle C(p_{1}\times p_{2})\geq C(p_{1})+C(p_{2})} . Let X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} be two independent random variables. Let Y 1 {\displaystyle Y_{1}} be
8514-629: The product channel, P ( Y 1 , Y 2 = y 1 , y 2 | X 1 , X 2 = x 1 , x 2 ) = P ( Y 1 = y 1 | X 1 = x 1 ) P ( Y 2 = y 2 | X 2 = x 2 ) {\displaystyle \mathbb {P} (Y_{1},Y_{2}=y_{1},y_{2}|X_{1},X_{2}=x_{1},x_{2})=\mathbb {P} (Y_{1}=y_{1}|X_{1}=x_{1})\mathbb {P} (Y_{2}=y_{2}|X_{2}=x_{2})} . For
8613-402: The random channel gain | h | 2 {\displaystyle |h|^{2}} , which is unknown to the transmitter. If the transmitter encodes data at rate R {\displaystyle R} [bits/s/Hz], there is a non-zero probability that the decoding error probability cannot be made arbitrarily small, in which case the system is said to be in outage. With
8712-431: The rate R {\displaystyle R} as: A rate is said to be achievable if there is a sequence of codes so that the maximum probability of error tends to zero as n {\displaystyle n} approaches infinity. The capacity C {\displaystyle C} is the highest achievable rate. Consider a codeword of length n {\displaystyle n} sent through
8811-399: The rate from the last section. A codebook, known to both encoder and decoder, is generated by selecting codewords of length n , i.i.d. Gaussian with variance P − ε {\displaystyle P-\varepsilon } and mean zero. For large n, the empirical variance of the codebook will be very close to the variance of its distribution, thereby avoiding violation of
8910-402: The received vector. Define the following three events: An error therefore occurs if U {\displaystyle U} , V {\displaystyle V} or any of the E i {\displaystyle E_{i}} occur. By the law of large numbers, P ( U ) {\displaystyle P(U)} goes to zero as n approaches infinity, and by
9009-488: The same kind of noise in communication systems, but described it in terms of frequencies. He described his findings to Harry Nyquist , also at Bell Labs, who used principles of thermodynamics and statistical mechanics to explain the results, published in 1928. Johnson's experiment (Figure 1) found that the thermal noise from a resistance R {\displaystyle R} at kelvin temperature T {\displaystyle T} and bandlimited to
9108-487: The single-antenna, point-to-point scenario. For channel capacity in systems with multiple antennas, see the article on MIMO . If the average received power is P ¯ {\displaystyle {\bar {P}}} [W], the total bandwidth is W {\displaystyle W} in Hertz, and the noise power spectral density is N 0 {\displaystyle N_{0}} [W/Hz],
9207-459: The terahertz, far beyond the capabilities of conventional electronics, and so it is valid to set η ( f ) = 1 {\displaystyle \eta (f)=1} for conventional electronics work. Nyquist's formula is essentially the same as that derived by Planck in 1901 for electromagnetic radiation of a blackbody in one dimension—i.e., it is the one-dimensional version of Planck's law of blackbody radiation . In other words,
9306-453: The thermodynamic distribution of the number of electrons on the capacitor, even without the involvement of a resistor. The noise is not caused by the capacitor itself, but by the thermodynamic fluctuations of the amount of charge on the capacitor. Once the capacitor is disconnected from a conducting circuit, the thermodynamic fluctuation is frozen at a random value with standard deviation as given above. The reset noise of capacitive sensors
9405-513: The total average power transferred over bandwidth Δ f {\displaystyle \Delta f} from R 1 {\displaystyle R_{1}} and absorbed by R 2 {\displaystyle R_{2}} was determined to be: Simple application of Ohm's law says the current from V 1 {\displaystyle V_{1}} (the thermal voltage noise of only R 1 {\displaystyle R_{1}} ) through
9504-682: The two resistors dissipates noise in both itself and in the other resistor. Since only half of the source voltage drops across any one of these resistors, this maximum noise power transfer is: This maximum is independent of the resistance and is called the available noise power from a resistor. Signal power is often measured in dBm ( decibels relative to 1 milliwatt ). Available noise power would thus be 10 log 10 ( k B T Δ f 1 mW ) {\displaystyle 10\ \log _{10}({\tfrac {k_{\text{B}}T\Delta f}{\text{1 mW}}})} in dBm. At room temperature (300 K),
9603-464: The uncertainty in the zero crossing. As the amplitude of the AWGN is increased, the signal-to-noise ratio decreases. This results in increased uncertainty Δ t . When affected by AWGN, the average number of either positive-going or negative-going zero crossings per second at the output of a narrow bandpass filter when the input is a sine wave is where In modern communication systems, bandlimited AWGN cannot be ignored. When modeling bandlimited AWGN in
9702-576: The underlying behavior of a system before these other phenomena are considered. The AWGN channel is a good model for many satellite and deep space communication links. It is not a good model for most terrestrial links because of multipath, terrain blocking, interference, etc. However, for terrestrial path modeling, AWGN is commonly used to simulate background noise of the channel under study, in addition to multipath, terrain blocking, interference, ground clutter and self interference that modern radio systems encounter in terrestrial operation. The AWGN channel
9801-457: Was the reason C.E. Shannon chose feedback as the subject of the first Shannon Lecture, delivered at the 1973 IEEE International Symposium on Information Theory in Ashkelon, Israel. The feedback capacity is characterized by the maximum of the directed information between the channel inputs and the channel outputs, where the maximization is with respect to the causal conditioning of the input given
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